OpenAI proves ω ≤ 9/4 for matrix multiplication

Your AI and this story

Key facts

• Theorem 1.1 (verbatim): ‘For every ε > 0, two n × n complex matrices can be multiplied using Oε(n^{9/4+ε}) arithmetic operations. In particular, ω ≤ 9/4.’ The paper adds that its proof ‘does not specify a competitive finite matrix size’. 13 pages, dated 2 Oct 2026
• Companions in the family: ‘Complex Matrix Multiplication Below 2.258 and Rectangular Bounds’ (83 pp., 24 Sep; ω < 2.258, dual exponent α > 0.465, ω(1, 0.709, 1) < 2.092 in characteristic zero) and ‘Staggered extraction for exact matrix multiplication over every field’ (35 pp., 24 Sep; ω < 2.371054886006746 over every fixed field, including positive characteristic)
• Method (paper): ‘The present proof combines explicit tensor degenerations with the spectral viewpoint.’ A separation construction turns blocks that share one tensor leg into independent blocks; two inequalities for polynomial-multiplication tensors force a growth rate a^{4/3}, which gives 3t ≤ 9/4 for every spectral character
• Previous records (as listed by Pass): ω < 2.371339 (Alman et al., 2025) and ω < 2.371177 (Dupont et al., August 2026, found with AlphaEvolve; see AlphaEvolve helps lower the matrix multiplication exponent ω to below 2.371177)
• Lean: comparator challenge MatrixMultiplication.lean defines straight-line programs with constant, input, add, subtract and multiply gates, counts the arithmetic gates, and defines ω as the infimum of exponents τ admitting correct programs of cost C·n^(τ+ε) for every n; theorem complex_omega_le_nine_quarters (plus α > 93/200 and the rectangular bound). The oai-math-revised comparator run (25–26 Oct) accepted it
• Amanov, van den Berg and Zuiddam (arXiv 2610.30054, 21 Oct, verbatim): ‘We give an exposition of OpenAI’s recent proof that the exponent ωC of matrix multiplication over the complex numbers satisfies ωC ≤ 9/4. Our version of the proof gives ωF ≤ 9/4 over every field F.’ They prove an ‘operational barrier’ showing the current analysis cannot be pushed further with the same ingredients, and note that Sela Navot, Kunal Marwaha and they themselves each released Lean proofs of ω_F ≤ 9/4 for every field on 7 Oct
• Rafael Pass (arXiv 2610.28627, 20 Oct, verbatim): ‘We give a self-contained constructive proof that the matrix multiplication exponent satisfies ω ≤ 9/4 … We explicitly construct a Strassen-style matrix multiplication identity and obtain the bound by ordinary block recursion.’ His AI statement says Codex (Astra) was used extensively, while ‘The main ideas behind the construction, analysis, and abstractions are my own’
• Other follow-ups: Przemysław Uznański (arXiv 2610.10494) extends the analysis to rectangular products, ω(1, 1/2, 1) = 2, and all-pairs shortest paths in directed unweighted graphs in O(n^2.5); Houxi Lü (arXiv 2610.18914) replaces a Fourier step by polynomial interpolation to get ω ≤ 9/4 in every characteristic